This Week at ICERM
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April 21, 2024
There are no events currently scheduled for April 21st.
April 22, 2024
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3:30 - 4:00 pm EDTCoffee Break11th Floor Collaborative Space
April 23, 2024
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12:00 - 1:00 pm EDTMultigrid Part 2Post Doc/Graduate Student Seminar - 11th Floor Conference Room
- Casey Cavanaugh, Louisiana State University
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3:30 - 4:00 pm EDTCoffee Break11th Floor Collaborative Space
April 24, 2024
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11:00 am - 12:00 pm EDTTaylor-Series Expansion for Meshfree Methods in Computational Solid Mechanics11th Floor Lecture Hall
- Yuri Bazilevs, Brown University
Abstract
Meshfree methods, such as the Reproducing Kernel Particle Method (RKPM) are entering their fourth decade of development. While a powerful concept, the ability of meshfree methods like RKPM to realize their full potential hinges on overcoming several technological challenges. One of them, and perhaps the thorniest of all, is the development of numerical integration or quadrature techniques that are stable, efficient, and accurate. In FEM, or in IGA, the decomposition of the problem domain into elements allows the analyst to develop quadrature techniques that are element-based. Because approximation functions are infinitely smooth over the element interiors, Gaussian quadrature presents a provably accurate solution that hinges on its ability to optimally integrate smooth functions. However, the meshfree nature of methods like RKPM makes the definition of integration zones somewhat ambiguous. In addition, the infinite smoothness property inside the integration zones is also lost. As a result, traditional quadrature methods do not present a good practical solution for RKPM and similar techniques.
Numerical quadrature based on Taylor-series expansion approaches was introduced in the mid-80s for FEM to develop a parameter-free approach for hourglass control. It laid dormant for a while, and, much later, in the mid 2010s, it resurfaced in the context of RKPM to develop a so-called Natural Stabilization approach, which is arguably the most important recent breakthrough in RKPM that brought the necessary added robustness for a wide range of nonlinear solid mechanics applications. In this talk, I will present a general framework of Taylor-expansion-based methods in computational solid mechanics and its broad applicability to meshfree methods and beyond. I will demonstrate: i. How to develop Taylor-series-expansion-based formulations that are accurate and stable for nearly incompressible deformations; ii. How to stabilize correspondence-based Peridynamics without resorting to costly bond-associated approaches; and iii. How to develop general-purpose large-deformation meshfree thin shells. -
3:30 - 4:00 pm EDTCoffee Break11th Floor Collaborative Space
April 25, 2024
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11:00 am - 12:00 pm EDTA quasi-incompressible Chan-Hilliard-Darcy model for two-phase flows in porous media11th Floor Lecture Hall
- Daozhi Han, State University of New York at Buffalo
Abstract
Two-phase flows in porous media is known as the Muskat problem. The Muskat problem can be ill-posed. In this talk we introduce a quasi-incompressible Cahn-Hilliard-Darcy model as a relaxation of the Muskat problem. We show global existence of weak solution to the model. We then present a high order accurate bound-preserving and unconditionally stable numerical method for solving the equations.
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3:30 - 4:00 pm EDTCoffee Break11th Floor Collaborative Space
April 26, 2024
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3:30 - 4:00 pm EDTCoffee Break11th Floor Collaborative Space
April 27, 2024
There are no events currently scheduled for April 27th.
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